Rebels near Misrata. The fighters are battle-hardened after defending their city from Gaddafi's forces for nearly four months. Photograph: Hassan Ammar/AP

Muammar Gaddafi's increasingly stretched army is struggling to crush a surprise rebel offensive in Zawiya, a strategic town just 30 miles west of Tripoli, while fighting has also been reported in the largest town in the Libyan Sahara.

At least 13 rebel fighters and civilians were killed in Zawiya on Saturday after the attack on government forces.

Guma el-Gamaty, a London-based opposition spokesman, said on Sunday several hundred rebels remained "well-protected and entrenched" in the western part of the town and also controlled a section of the coastal highway to Tunisia that serves as the main supply route for Tripoli. Loyalists forces were still in charge of the town centre and have large numbers of snipers there, he said by phone from London.

The government acknowledged that fighting had occurred in Zawiya, which has a major oil refinery, but said the clashes were minor and loyalist forces remained in complete control.

It was not possible to verify the conflicting information since foreign journalists in Tripoli are not permitted to leave the capital.

Zawiya first rebelled against the Libyan regime in March, but the uprising was crushed when the army sent in tanks and bulldozers, even razing the town's mosque. If sustained, the town's second rebellion will be of major significance, because it will mean that Gaddafi's forces are engaged in fighting in all three directions outside the capital, including Misrata, 130 miles to the east and a string of towns in the Nafusa mountains, about 60 miles to the south.

"The strategy is to push on Tripoli from all three sides, inspiring people there and putting great pressure on the army," said el-Gamaty, the UK coordinator for the rebel's National Transitional Council (NTC).

A businessman in Tripoli, who requested anonymity for his own safety, said the strain on the regime appeared to be showing, with security tightening sharply in Tripoli in recent days. Vehicles entering suburbs known to be restive at night were being checked by up to six times at a single roadblock by police officers, the businessman said. Gaddafi remains in hiding in the city with his family, and appears not to be responding to international diplomatic efforts offering him a safe exit from power.

However Libyan state television on Sunday broadcast pictures of Gaddafi meeting the president of the international chess federation.

Russia's Interfax news agency quoted Kirsan Ilyumzhinov, who is also a former Russian provincial governor, as saying he played a game of chess in Tripoli on Sunday with Gaddafi.

He said the Libyan leader told him he had no intention of leaving the country.

Following a visit to Benghazi, the headquarters of the NTC in eastern Libya, Russia's special envoy to Africa, Mikhail Margelov told Russian television that the "clock was ticking" for Gaddafi, but said there were precedents for a solution where the Libyan leader might be able "to live happily as a private individual in his own country and nobody touches him". Libyan state television broadcast pictures of Gaddafi meeting the president of the international chess federation.

Russia's Interfax news agency quoted Kirsan Ilyumzhinov, who is also a former Russian provincial governor, as saying he played a game of chess in Tripoli with Gaddafi.

Besides Zawiya, the NTC also reported skirmishes in parts of Sahba, a garrison town around 500 miles south of Tripoli that is considered a Gaddafi stronghold. Anti-government protests have reportedly been ongoing there for several days. Regarded as the gateway to the Sahara, Sabha is also a key junction for routes to Algeria, Niger and Chad that could be used by senior regime officials looking to escape the country, according to rebels.

They remain hopeful that Nato's aerial operation targeting Gaddafi's military arsenal might wear it down to the extent that rebel fighters may soon be able to push through government defences on the various fronts. If and when that happens, the main thrust on Tripoli is likely to come from the rebels in Misrata, Libya's third biggest city, perhaps reinforced by a larger opposition force currently holding its position hundreds of miles east near the town of Brega, which is still in government hands.

Misrata's fighters are battle-hardened after defending their city for nearly four months, and commanders claim that increasingly numbers of government troops are deserting. But they are aware of the shortcomings of their all-infantry force against the heavy weapons of loyalist forces. In an artillery barrage on Friday, the rebels lost 31 men, and suffered 120 injuries. Rebels say the 32nd Brigade, an elite force reportedly controlled by Gaddafi's son, Khamis, is garrisoning Zlitan, the town blocking any advance from Misrata on Tripoli.

Meanwhile, in the Nafusa mountain range to the south of Tripoli, government forces are continuing their attacks on the mainly Berber communities who have taken control of numerous small towns near the road that runs towards Tunisia. In the rebel-controlled town of Zintan, five people were killed during rocket attacks on Sunday, according to a rebel spokesman. More than 20 people were injured.

Sign up for the Guardian Today

Our editors' picks for the day's top news and commentary delivered to your inbox each morning.